# Engineering Acoustics/Forced Oscillations (Simple Spring-Mass System)

### Spring-mass system

Spring-mass system in equilibrium (A), compressed (B) and stretched (C) states.

When a spring is stretched or compressed by a mass, the spring develops a restoring force. Hooke's law gives the relationship of the force exerted by the spring when the spring is compressed or stretched a certain length:

$F \left( t \right) =-kx \left( t \right)$

where F is the force, k is the spring constant, and x is the displacement of the mass with respect to the equilibrium position.

This relationship shows that the distance of the spring is always opposite to the force of the spring.

By using either force balance or an energy method, it can be readily shown that the motion of this system is given by the following differential equation:

$F(t) = -kx(t) = m \frac {\mathrm{d}^{2}}{\mathrm{d}{t}^{2}} x \left( t \right) = ma.$

...the latter evidently being Newton's second law of motion.

If the initial displacement is A, and there is no initial velocity, the solution of this equation is given by:

$x \left( t \right) =A \cos \left( \sqrt{k \over m}t \right).$

Given an ideal massless spring, $m$ is the mass on the end of the spring. If the spring itself has mass, its effective mass must be included in $m$.

#### Energy variation in the spring-damper system

In terms of energy, all systems have two types of energy, potential energy and kinetic energy. When a spring is stretched or compressed, it stores elastic potential energy, which then is transferred into kinetic energy. The potential energy within a spring is determined by the equation $U = k{x}^{2}/ 2 .$

When the spring is stretched or compressed, kinetic energy of the mass gets converted into potential energy of the spring. By conservation of energy, assuming the datum is defined at the equilibrium position, when the spring reaches its maximum potential energy, the kinetic energy of the mass is zero. When the spring is released, the spring will try to reach back to equilibrium, and all its potential energy is converted into kinetic energy of the mass.